Upper bounds for uniform Lebesgue constants of interpolation periodic sourcewise representable splines

A method for the construction of biorthogonal bases of multiwavelets from known bases of multiscaling functions is given. It is similar to the method presented in the author’s 2014 paper joint with N.I. Chernykh and is based on the same principle: the construction of multiwavelets based on k multiscaling functions employs an analog of the vector product of vectors in a 2k-dimensional space.     

Hermite spline interpolation in the discrete periodic case

Interpolation of discrete periodic complex-valued functions by the values and increments given at equidistant nodes is examined. A space of discrete functions in which the interpolation problem is uniquely solvable is introduced. Extremal and limit properties of the solution to this problem are found.     

An efficient algorithm for periodic Hermite spline interpolation with shifted nodes

Generalized Hermite spline interpolation with periodic splines of defect 2 on an equidistant lattice is considered. Then the classic periodic Hermite spline interpolation with shifted interpolation nodes is obtained as a special case.By means of a new generalization of Euler-Frobenius polynomials the symbol of the considered interpolation problem is defined. Using this symbol, a simple representation of the fundamental splines can be given. Furthermore, an efficient algorithm for the computation of the Hermite spline interpolant is obtained, which is mainly based on the fast Fourier transform.     

Quadratic spline interpolation on uniform meshes

The interpolation problem at uniform mesh points of a quadratic splines(x _i)=f _i,i=0, 1,...,N ands(x ₀)=f₀ is considered. It is known that s–f=O(h ³) and s–f=O(h ²), whereh is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent off other than mesh points wheres=f by assumption. Best error bounds for some compound formulae approximatingf _i andf _i ⁽³⁾ are also derived.     

Lebesgue constants of Gaussian cardinal interpolation operators from l (ℤ) into L (ℝ)

Summary In recent years, with the attention to the radial-basis function by mathematicians, more and more research is concentrated on the Gaussian cardinal interpolation. The main purpose of this paper is to discuss the asymptotic behavior of Lebesgue constants of the Gaussian cardinal interpolation operator ℒ_λ from l (ℤ) into L (ℝ), that is, ∥ℒ_λ∥₁. We obtain the strong asymptotic estimate of the Lebesgue constants which improves the results of Riemenschneider and Sivakumar in [11].     

On the interpolation constants for variable Lebesgue spaces

For $\theta \in (0,1)$ and variable exponents $p_0(·),q_0(·)$ and $p_1(·),q_1(·)$ with values in [1, ∞], let the variable exponents $p_{\theta }(·),q_{\theta }(·)$ be defined by $$1/p_{\theta }(·):=(1-\theta )/p_0(·)+\theta /p_1(·),1/q_{\theta }(·):=(1-\theta )/q_0(·)+\theta /q_1(·).$$ The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space $L^{p_j(·)}$ to the variable Lebesgue space $L^{q_j(·)}$ for $j=0,1$, then $${\parallel T\parallel }_{L^{p_{\theta }(·)}\to L^{q_{\theta }(·)}}\le {C\parallel T\parallel }_{L^{p_0(·)}\to L^{q_0(·)}}^{1-\theta }{\parallel T\parallel }_{L^{p_1(·)}\to L^{q_1(·)}}^{\theta },$$ where C is an interpolation constant independent of T. We consider two different modulars ${\varrho }^{\max }(·)$ and ${\varrho }^{\mathrm{sum}}(·)$ generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants C_max and C_sum, which imply that $C_{\max }\le 2$ and $C_{\mathrm{sum}}\le 4$, as well as, lead to sufficient conditions for $C_{\max }=1$ and $C_{\mathrm{sum}}=1$. We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that $p_j(·)=q_j(·)$, $j=0,1$ are Lipschitz continuous and bounded away from one and infinity (in this case, ${\varrho }^{\max }(·)={\varrho }^{\mathrm{sum}}(·)$).     

Quasi-interpolants from spline interpolation operators

For bi-infinite Toeplitz matrices, it is easy to see that thekth partial sum of the Neumann series reproduces polynomials of orderk There is no guarantee, however, that the spectral radius is less than 1. A principal result of this paper is to show that for the spline interpolation Toeplitz case the spectral radius is less than 1 whenA is invertible and the main diagonal is the central diagonal. This is not true for all totally positive Toeplitz matrices as shown by an example in Section 2.Communicated by Charles A. Micchelli.     

Optimal shift parameters for periodic spline interpolation

Using the exponential Euler spline, restricted on the unit circle, we sketch a unified approach to the periodic spline interpolation with shifted interpolation nodes. Mainly we are interested in the optimal choice of the shift parameter such that the corresponding interpolatory matrix possesses minimal condition or such that the related interpolation operator has minimal norm. We show that =0 is optimal in both cases. This improves known results of Merz, Reimer-Siepmann and Richards.     

On the Lebesgue constants for interpolation of analytic functions

Дль сИстЕМы РАжлИЧНы х тОЧЕкΤ=(t ₁,...,t _n ) Иж ОтРЕ жкА [?1,1] Иk?[0,1) ВВОДИтсь ВЕлИЧ ИНА $$L_n (\tau ,p,k) = \mathop {\max }\limits_{t \in [ - 1,1]} (\mathop \Sigma \limits_{j = 1}^n |D_j (t)|^p )^{1/p} ,$$ где $$D_j (t) = \frac{{\omega _j (t)}}{{\omega _j (t_j )}}[1 - kW_j^2 (t)],{\mathbf{ }}\omega _j (t) = \mathop \prod \limits_{\begin{array}{*{20}c} {m = 1} \\ {m \ne 1} \\ \end{array} }^n W_m (t),{\mathbf{ }}W_m (t) = \frac{{t - t_m }}{{1 - kt_m t}}.$$ пРИk=0 ОНА сОВпАДАЕт с кОНс тАНтОИ лЕБЕгА, сВьжАН НОИ с ИНтЕРпОльцИЕИ МНОгО ЧлЕНОМ лАгРАНжА. пОкАжАНА сВ ьжь ВЕлИЧИНыL _n (Τ, p, k) с жАД АЧАМИ ИНтЕРпОльцИИ АНАлИт ИЧЕскИх ФУНкцИИ. Дль сИстЕМы $$Z = \left\{ {sn\left[ {\left( {\frac{{2j - 1}}{n} - 1} \right)K,k} \right]} \right\}_{j = 1}^n ,$$ ьВльУЩЕИсь АНАлОгОМ ЧЕБышЕВскОИ сИстЕМы, пОлУЧЕНы ОцЕНкИL _n (Z, p, k) пРИp≧2 Иp≧1.     

Cardinal interpolation and spline fucntions V. The B-splines for cardinal Hermite interpolation

In the third paper of this series on cardinal spline interpolation [4] Lipow and Schoenberg study the problem of Hermite interpolation

$\text{S(v) = Y}{}_{\mathrm{v}}\text{, S′(v) = Y}{}_{\mathrm{v}}\text{′,…,S}{}^{\mathrm{(r?1)}}\text{(v) = Y}{}_{\mathrm{v}}{}^{\mathrm{(r?1)}}\text{for all}\text{v}$

. The B-splines are there conspicuous by their absence, although they were found very useful for the case γ = 1 of ordinary (or Lagrange) interpolation (see [5–10]). The purpose of the present paper is to investigate the B-splines for the case of Hermite interpolation (γ > 1). In this sense the present paper is a supplement to [4] and is based on its results. This is done in Part I. Part II is devoted to the special case when we want to solve the problem

$\text{S(v) = Y}{}_{\mathrm{v}}\text{, S′(v) = Y}{}_{\mathrm{v}}\text{′}\text{for all}\text{v}$

by quintic spline functions of the class C?(– ∞, ∞). This is the simplest nontrivial example for the general theory. In Part II we derive an explicit solution for the problem (1), where v = 0, 1,…, n.     

The possibility of periodic spline interpolation

A new case of the solvability of the classical interpolation problem for periodic splines is described.Translated from Matematicheskie Zametki, Vol. 8, No. 5, pp. 563–573, November, 1970.I wish to thank S. B. Stechkin for his valuable suggestions and constant interest in this work.     

Uniform bounds for cardinal Hermite spline operators with double knots

A method is given for computing the uniform norm of the cardinal Hermite spline operator. This is the operator that takes two bounded biinfinite sequences of numbers into the unique bounded spline of degree 2k − 1(k 2) with knots of multiplicity two at the integers and that interpolates the two given sequences for both functional and first derivative values at the integers. The computational schema relies on knowledge of the Bernoulli splines, while the theoretical aspects make use of some properties of zeros of periodic splines.     

Error bounds in periodic quartic spline interpolation

In this paper we develop periodic quartic spline inter polation theory which, in general, gives better fits to continuous functions than does the existing quintic spline inter polation theory. The main theorem of the paper is to establish that ⋎s^(r)-y^(r)⋎=O(h^6−r), r=0,1,2,3. Also, the nonperiodic cases cannot be constructed empolying the methodology of this paper because that will involve several other end conditions entirely different than (1.10).     

Complex interpolation and regular operators between Banach lattices

Supported in part by N.S.F. grant DMS 9003550.     

On uniform Lebesgue constants of local exponential splines with equidistant knots

For a linear differential operator L _r of arbitrary order r with constant coefficients and real pairwise different roots of the characteristic polynomial, we study Lebesgue constants (the norms of linear operators from C to C) of local exponential splines corresponding to this operator with a uniform arrangement of knots; such splines were constructed by the authors in earlier papers. In particular, for the third-order operator L ₃ = D(D ² ? β ²) (β > 0), we find the exact values of Lebesgue constants for two types of local splines and compare these values with Lebesgue constants of exponential interpolation splines.     

On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at −∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(⋅)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I−S)/2 is Fredholm on the variable Lebesgue space , then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces and with some exponents ql and qr lying in the segments between the lower and the upper limits of p at −∞ and +∞, respectively.